Many of our results need the assumption that \(C^1(\Omega )\) -functions are dense in \(W^{1,\varphi }(\Omega )\) .
A suuficient condition is given by [1]}, which states that \(C^\infty (\Omega )\) -functions are dense in \(W^{1,\varphi }(\Omega )\) , if \(\varphi \)  satisfies (A0), (A1), (A2) and (aDec).
By [1]}, (A2) can be omitted, if \(\Omega \)  is bounded.
